Real-rooted atlas

A038221 → basis A013611 → basis A008277

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(18 + 12\,x\right)\,P_{n-1}(x) + \left(12\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1181230
23245761441044
358322116812960172841688
4104976705024798336248832207361877904
51889568224104324199040023016960435456024883293910752
634012224694987776203252198417468006405636044807166361629859845146576704
761222003221257779968937057766401187174983685815950336012353015808112870195235831808305970327936
81101996057664508003942441907359784967535214231552526313720217616614492733442498193653761719926784042998169619574085300480
91983592903681948464070963218375187156992045719040024576043778962612224019275068974694442421134163968475215770419225540912742451597803521338599448460800
103570467226624586919532773376795139429445222426893304072110080343804001270169602028802931372851261309103517204489937736957952008614253297664037150418534406191736422497318221347017728
116426841007923217650431589920768341001594760273920154765515142737100825918260572484403201995031087995027456795238283419582464174935841191755776217455331088793601501805669253120531250985041927430083706887486703922927458304
121156831381426176530284168618573824145338721590105538568766539731420132147218963238159552551321618671433755059342540895511818563702751232273863152025940787204578220868713316352450173911632445440253529316246159367489524376535048916100448256607030068966136123392
1320822964865671168159224070351343288326167860407018661478404908337920874998005760135677219530750997299201683930690809259334041610836635721340368715776393954597342902904422485055296146444700876811166056329024792166489007915037863444484165780451434168321043183752445952010699320537907251695819180648309514240
143748133675820810244779220866324179189762609608259389999015526427245168021567228608512095436970395345655824384014762786022681234773114881177368897817744304504832531873317945845387100160144237589288095307530240242758665121446022348802568235987005490593792169296204816948658176668889422068344422414379886802947276812839184645488644610174403663811529883648
Technical details
Node id
n-8ed7c3031d730586 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-5162632e955115eb

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:            1;
  1:           18,             12;
  2:          324,            576,             144;
  3:         5832,          21168,           12960,            1728;
  4:       104976,         705024,          798336,          248832,           20736;
  5:      1889568,       22410432,        41990400,        23016960,         4354560,          248832;
  6:     34012224,      694987776,      2032521984,      1746800640,       563604480,        71663616,        2985984;
  7:    612220032,    21257779968,     93705776640,    118717498368,     58159503360,     12353015808,     1128701952,      35831808;
  8:  11019960576,   645080039424,   4190735978496,   7535214231552,   5263137202176,   1661449273344,   249819365376,   17199267840,    429981696;
  9: 198359290368, 19484640709632, 183751871569920, 457190400245760, 437789626122240, 192750689746944, 42421134163968, 4752157704192, 255409127424, 5159780352;
%F Formula
The row polynomials satisfy P(n,x) = (18+12*x)*P(n-1,x) + 12*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(18 + 12*x)*P[n - 1, x] + 12*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (18 + 12*x)*v[i - 1] + 12*x*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

Proof chains ? JSON

Same fingerprint ?

References

  1. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  7. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  8. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.